
One first shows that there is a geometric \(U\) -quotient \(q_U: Y_{\min } \rightarrow Y_{\min }/U\)  by gluing together affine geometric quotients; this uses the grading 1-PS \(\lambda \)  and the fact that REF  holds (see [1]}).

Then the grading 1-PS \(\lambda \)  is also used to give a projective \(\widehat{U}\) -quotient as follows. First one constructs a \(\lambda (\mathbb {G}_m)\) -equivariant embedding of \(Y_{\min }/U\)  in a projective space and we let \(\overline{Y_{\min }/U}\)  denote its projective completion (see [1]}). Then the projective \(\widehat{U}\) -quotient is constructed by taking a reductive quotient of \(\overline{Y_{\min }/U}\)  by \(\lambda (\mathbb {G}_m)\) , where one slightly perturbs the adapted linearisation by twisting by a rational character to obtain a so-called well-adapted linearisation (cf. [1]}).

If REF  holds, then by [4]}, \(Y^{\widehat{U}-s}\)  coincides with the locally trivial stable locus (see [4]}, and it is shown that this geometric \(\widehat{U}\) -quotient is projective and coincides with the enveloping quotient (see [4]}). Hence, the ring of \(\widehat{U}\) -invariant sections of a sufficiently divisible tensor power of this well-adapted linearisation is finitely generated and its projective spectrum is \(Y^{\widehat{U}-s}/\widehat{U}=Y/\!/\widehat{U}\) . In particular, \(q_{\widehat{U}}\)  can be constructed by taking invariant sections for a slight perturbation of the original linearisation.

 In general we will assume \(U Z_{\min } \ne Y_{\min }\) . If this does not hold then the \(\widehat{U}\) -stable set \(Y^{\widehat{U}-s}\)  is empty and in this case, we should define \(Y_{\min }\)  as a replacement stable set which admits a the geometric \(p: Y_{\min } \rightarrow Z_{\min }\)  (see [1]}). Moreover, there is a good \(H\) -quotient \(UZ_{\min }^{ss} \stackrel{p}{\rightarrow } Z_{\min }^{ss} \rightarrow Z_{\min } /\!/_{\! \mathcal {L}} \overline{L}\) .


